Theorems of the alternative for inequality systems of real polynomials
This paper presents two new results on the matrix decomposition. One of them is given under an assumption proposed by Ai, Huang, and Zhang in the context of the complex space and the other result is given under a new assumption that at least one of the three involved matrices is negative semidefinite. The authors show that several theorems of the alternative are true under the new proposed assumption. The inequality system of higher degree real polynomials defined by even order tensors is also considered in this paper since it appears in many applied science areas. A theorem of the alternative for the inequality system of two higher degree real polynomials is established under suitable assumptions. In addition, as an application of the obtained theorem of the alternative, the authors investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions and show that the extreme \(\mathbb{Z}\)-eigenvalue problem is a special case of this class of polynomial optimization problems.
scientific article; zbMATH DE number 6110391
scientific article